If you're just starting to play the game, chances are you were asked if you wanted to play on EASY. The problem with this would be the fact that S rank is impossible to get in Legend Log with this option toggled. Therefore, you have two options: playing Legend Log on Normal (or Hard, but not advised until you start getting some higher level characters, maybe even level break characters), or play Dream Log. If you've at least cleared the first chapter, one character should be more than enough to sweep through the first island in Dream Log.
The evaluation is determined by three things: kills, beats ("!"), and time taken to complete. Typically, the more you kill, the more beat you acquire, and the quicker you complete the mission, the better your chances of clearing with an S rank. I do believe, though, that this would be easier in Dream Log simply because there appears to be a rank gauge in the lower right (above "!" and kills) that changes based on the kills and beat you end up gathering (of course, this may change based on the time you take too, but I've never wasted time to check and find out). For this reason I would suggest Dream Log's first island, if you're playing through on Easy your first time in Legend Log to unlock characters and get your basic level built up on one character (so you can Beli boost your other characters later).
General S-rank requirements for the Dream Log islands:
2000 Kills
1000 "!" (Beat)
Clearing in under 18 minutes